Research on the innovation ability evaluation of traditional enterprise’s business model for internet transition with hesitant fuzzy information

Abstract

Since 1990 s, the Internet as the representative’s information technology fast development, has provided richer innovation way and the innovation space for enterprises compared to formerly any time all. Many new pattern enterprises arise at the historic moment based on the Internet technology. They create and provide the value with the traditional enterprise completely different way to the customer, and obtained huge success that the traditional enterprise to be unable. Under such background, business model–this beginning be seen in the computer specialized glossary, starts to arouse the social widespread interest, and becomes one of the most popular glossaries in academic circles in the worldwide scale in very short time. Now, the business model and the business model innovation in the global scope, brings to the unprecedented attention and the widespread application. Even so, some very important concepts and fundamental theories about the business model fundamental research, such as the concept of business model, the factors that influence the innovation of business model and so on, do not form the uniform conclusion which accepts generally in the theorists. All of that on objective, has formed certain adverse effect to the business model innovation practice, therefore, the fundamental research related to business model and the innovation of business model urgently awaits to strengthen. In this paper, we investigate the multiple attribute decision making with hesitant fuzzy information. Motivated by the induced Choquet ordered averaging operator (I-COA) operator and geometric mean, we develop the induced hesitant fuzzy Hamacher correlated geometric (IHFHCG) operator and then utilize IHFHCG operator to develop the model for multiple attribute decision making with hesitant fuzzy information. Finally, an illustrative example for evaluating the innovation ability of traditional enterprise’s business model for internet transition is given to verify the developed approach.

Keywords

Multiple attribute decision making hesitant fuzzy information induced hesitant fuzzy Hamacher correlated geometric (IHFHCG) operator induced Choquet ordered averaging operator (I-COA) operator innovation ability of traditional enterprise’s business model

1 Introduction

The large scale state-owned enterprises and state-owned stock assets, especially including iron and melt industrial enterprises, raw materials manufacturing industrial enterprises and extractive industrial enterprises, which were built up in the period of planned economic system, made enormous contribution for the development of national economy and form the important foundation of industrial system of our country. However, during the process of going through transition in economic system, the competition advantage that they once had is being lost constantly, but the modern advantage with required market economy has not been set up. In these enterprises, institution is incomplete, procedure is slow and organization is machinery.Their lack spontaneous ability and gradually become the important burden of economic development. And, compared with small and medium-sized state-owned enterprises, it is the most outstanding and centralized that the drawback and mark of the old system in the state-owned large scale enterprises, and the degree of difficulty of the reform is greater, too. During the process of going through the period of system transition, if the huge stock assets can’t realize turnaround, the setting-up of the market economic system of our country and choice of the industrialized route, the micro foundation is not firm. Traditional advantage enterprises are large-scale state-owned enterprises which consumed resources and had certain advantage effectively under traditional planned economic system (including state holding enterprises).Looking from time, they were built up in traditional planned economy period mainly (“First Five-Year Plan” periodand “the strategic hinterland of China build” period especially), which had not stronger competitiveness until the initial stage 1990 s; Looking from investors, they were mainly invested by the government, including central government and local government; Looking from organization form, most enterprises adopt traditional department layers of bureaucrat organization, which are rigidity still; Looking from industrial fields, it includes the enterprises of traditional industrial fields, such as metallurgy, Petroleum and Chemical industry, ordinary machine- building, textile, food, etc; Looking from to space distribution, they concentrate on being distributed in the Northeast and central and west regions and become important micro foundation and carrier of the old industrial base. Whether an enterprise is a traditional advantage enterprise, it can be judged through standard of time, standard of main part of property right, standard of scope of the enterprise and standard of industrial fields. The choice of the industrialized route of backward areas in China and emergence of the traditional old industrial base, need to use a kind of brand-new train of thought and method, namely need to focus on the stock assets of these traditional advantage enterprises closely and appraise again, especially the fixed assets. According to the assets characteristic of enterprises, we should appraise, analyze and excavate the enterprises’ development potentiality. Trough system turnaround, process reengineering, organization turnaround, industry turnaround and policies turnaround, we make them turnaround from traditional advantage to modern advantage and from external competition advantage to endogenous competition advantage. We should make great efforts to stock assets, drive increment assets with stock assets, drive stock assets with increment assets. By this we hope they can make greater contributions to national economic growth, increase of social employment and the persons who are correlated with of the interests.

In this paper, we investigate the multiple attribute decision making with hesitant fuzzy information [1 –17]. Motivated by the induced Choquet ordered averaging operator (I-COA) operator and geometric mean, we develop the induced hesitant fuzzy Hamacher correlated geometric (IHFHCG) operator and then utilize IHFHCG operator to develop the model for multiple attribute decision making with hesitant fuzzy information. Finally, an illustrative example for evaluating the innovation ability of traditional enterprise’s business model for internet transition is given to verify the developed approach.

2 Preliminaries

Torra [18] proposed the hesitant fuzzy set which permits the membership having a set of possible values. Xia and Xu [19] proposed some hesitant fuzzy information aggregation operators and their application to multiple attribute decision making. Xu and Xia [20, 21] developed the distance and correlation measures with hesitant fuzzy information.

Definition 1. [18] Given a fixed set X, then a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0, 1]. To be easily understood, Xia and Xu [19] express the HFS by mathematical symbol: $E = (〈 x, h_{E} (x) 〉 | x \in X),$ (1) where h_E (x) is a set of some values in [0, 1], denoting the possible membership degree of the element x ∈ X to the set E. For convenience, Xia and Xu [19] call h = h_E (x) a hesitant fuzzy element(HFE) and H the set of all HFEs.

Definition 2. [19] For a HFE h, $s (h) = \frac{1}{# h} \sum_{γ \in h} γ$ is called the score function of h, where # h is the number of the elements in h. For two HFEs h₁ and h₂, if s (h₁) > s (h₂), then h₁ > h₂; if s (h₁) = s (h₂), then h₁ = h₂. Based on the relationship between the HFEs and IFVs, Xia and Xu [19] define some new operations on the HFEs h, h₁ and h₂: $\begin{matrix} (1) h^{λ} = \cup_{γ \in h} {γ^{λ}}; \\ (2) λ h = \cup_{γ \in h} {1 - {(1 - γ)}^{λ}}; \\ (3) h_{1} \oplus h_{2} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {γ_{1} + γ_{2} - γ_{1} γ_{2}}; \\ (4) h_{1} \otimes h_{2} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {γ_{1} γ_{2}} . \end{matrix}$

T-norm and t-conorm are an important notion in fuzzy set theory, which are used to define a generalized union and intersection of fuzzy sets [22]. Hamacher product ⊗ is a t-norm and Hamacher sum ⊕ is a t-conorm, where $\begin{matrix} T (a, b) = a \otimes b = \frac{ab}{γ + (1 - γ) (a + b - ab)} \\ T * (a, b) = a \oplus b = \frac{a + b - ab - (1 - γ) ab}{1 - (1 - γ) ab} \end{matrix}$

Motivated by the Hamacher aggregation operators, the product ⊗ and the Hamacher sum ⊕, then the generalized intersection and union on two HFEs h₁ and h₂ become the Hamacher product(denoted by h₁ ⊗ h₂) and Hamacher sum(denoted by h₁ ⊕ h₂) of two HFEs h₁ and h₂, respectively, as follows [23]: $\begin{matrix} (1) h_{1} \otimes h_{2} \\ = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {\frac{γ_{1} γ_{2}}{γ + (1 - γ) (γ_{1} + γ_{2} - γ_{1} γ_{2})}} . \\ (2) {(h_{1})}^{λ} = \cup_{{γ_{1} \in h}_{1}} \\ {\frac{γ {(γ_{1})}^{λ}}{{(1 + (γ - 1) (1 - γ_{1}))}^{λ} + (γ - 1) {(γ_{1})}^{λ}}}, λ > 0 . \end{matrix}$

3 IHFHCG operators

Definition 3. [24] Let f be a positive real-valued function on X and m be a fuzzy measure on X. The induced Choquet ordered averaging operator of dimension n is a function I - COA

(R⁺ × R⁺) → R⁺, which is defined to aggregate the set of second argument of a list of 2-tuples (〈 u₁, f₁ 〉 , 〈 u₂, f₂ 〉 , ⋯ , 〈 u_n, f_n 〉) according to the following expression: $\begin{matrix} {I - COA}_{m} (〈 u_{1}, f_{1} 〉, 〈 u_{2}, f_{2} 〉, \dots, 〈 u_{n}, f_{n} 〉) \\ = \sum_{j = 1}^{n} f_{σ (j)} [m (A_{σ (j)}) - m (A_{σ (j - 1)})] \end{matrix}$ (2) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that u_σ(i-1) ≥ u_σ(i) for all j = 2, ⋯ , n, i.e., 〈u_σ(j), f_σ(j)〉 is the 2-tuple with u_σ(j) the jth largest value in the set (u₁, u₂, ⋯ , u_n), A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

In the following, we shall develop the induced hesitant fuzzy Hamacher correlated geometric (IHFHCG) operator based on the I-COA operator [24] and geometric mean [25, 26].

Definition 4. Let 〈u_j, h_j 〉 (j = 1, 2, ⋯ , n) be a collection of 2-tuples on X, and μ be a fuzzy measure on X, then we define the induced hesitant fuzzy Hamacher correlated geometric (IHFHCG) operator as follows: $\begin{matrix} {IHFHCG}_{μ} (〈 u_{1}, h_{1} 〉, 〈 u_{2}, h_{2} 〉, \dots, 〈 u_{n}, h_{n} 〉) \\ = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} \end{matrix}$ (3) where h_σ(j) is the h_j value of the IHFHCG pair 〈u_i, h_i〉 having the jth largest u_i (u_i ∈ [0, 1]), and u_i in 〈u_i, h_i〉 is referred to as the order inducing variable and h_i as the hesitant fuzzy arguments, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

Based on the operations of the hesitant fuzzy values described, we can drive the Theorem 6.

Theorem 5.Let 〈u_j, h_j 〉 (j = 1, 2, ⋯ , n) be a collection of 2-tuples onX, andμbe a fuzzy measure onX, then their aggregated value by using the IHFHCG operator is also a hesitant fuzzy variables, and

$\begin{matrix} {IHFHCG}_{μ} (〈 u_{1}, h_{1} 〉, 〈 u_{2}, h_{2} 〉, \dots, 〈 u_{n}, h_{n} 〉) = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} \\ = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\frac{2 \prod_{j = 1}^{n} γ_{σ (j)}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}{\prod_{j = 1}^{n} {(2 - γ_{σ (j)})}^{w {(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}_{j}} + \prod_{j = 1}^{n} {(γ_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}} \end{matrix}$ (4)

where h_σ(j) is the h_j value of the IHFHCG pair 〈u_i, h_i〉 having the jth largest u_i (u_i ∈ [0, 1]), and u_i in 〈u_i, h_i〉 is referred to as the order inducing variable and h_i as the hesitant fuzzy arguments, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

Now we consider some special cases of the IHFHCG operator:

(1) If u_j = h_j for all j, then the IHFHCG operator becomes the hesitant fuzzy Hamacher correlated geometric (HFHCG) operator:

$\begin{matrix} {IHFHCG}_{μ} (〈 u_{1}, h_{1} 〉, 〈 u_{2}, h_{2} 〉, \dots, 〈 u_{n}, h_{n} 〉) \\ = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} {= HFHCG}_{w} (h_{1}, h_{2}, \dots, h_{n}) \\ = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} = \cup_{γ_{σ (1)} \in h_{σ (1)}, γ_{σ (2)} \in h_{σ (2)}, \dots, γ_{σ (n)} \in h_{σ (n)}} \\ {\frac{2 \prod_{j = 1}^{n} γ_{σ (j)}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}{\prod_{j = 1}^{n} {(2 - γ_{σ (j)})}^{w {(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}_{j}} + \prod_{j = 1}^{n} {(γ_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))}}} \end{matrix}$ (5)

where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that h_σ(j-1) ≥ h_σ(j) for all j = 2, ⋯ , n.

(2) If u_j = No . j for all j, where j is the ordered position of the 〈u_j, h_j〉, then the IHFHCG operator becomes the hesitant fuzzy Hamacher weighted geometric (HFHWG) operator:

$\begin{matrix} {IHFHCG}_{μ} (〈 u_{1}, h_{1} 〉, 〈 u_{2}, h_{2} 〉, \dots, 〈 u_{n}, h_{n} 〉) \\ = \otimes_{j = 1}^{n} {(h_{σ (j)})}^{(μ (A_{σ (j)}) - μ (A_{σ (j - 1)}))} \\ = {HFHWG}_{ω} (h_{1}, h_{2}, \dots, h_{n}) \\ = \otimes_{j = 1}^{n} {(h_{j})}^{ω_{j}} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots, γ_{n} \in h_{n}} \\ {\frac{γ \prod_{j = 1}^{n} γ_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - γ_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} γ_{j}^{ω_{j}}}} \end{matrix}$ (6)

where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

4 Approaches to hesitant fuzzy multiple attribute decision making

The following assumptions or notations are used to represent the MADM problems with hesitant fuzzy

information. Let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives and G ={ G₁, G₂, ⋯ , G_n } be a set of attributes. If the decision makers provide several values for the alternative A_i under the state of nature G_j with anonymity, these values can be considered as a hesitant fuzzy element h_ij. In the case where two decision makers provide the same value, then the value emerges only once in h_ij. Suppose that the decision matrix H = (h_ij) _m×n is the hesitant fuzzy decision matrix, where h_ij (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n) are in the form of HFEs.

Based on the above models, we develop a practical method for solving the hesitant fuzzy MADM problems based on the IHFHCG operator. The method involves the following steps:

Step 1. Utilize the IHFHCG operator:

$\begin{matrix} h_{i} {= IHFHCG}_{μ} (〈 u_{i 1}, h_{i 1} 〉, 〈 u_{i 2}, h_{i 2} 〉, \dots, 〈 u_{in}, h_{in} 〉) \\ = \otimes_{j = 1}^{n} {(h_{σ (ij)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))} = \cup_{γ_{σ (i 1)} \in h_{σ (i 1)}, γ_{σ (i 2)} \in h_{σ (i 2)}, \dots, γ_{σ (in)} \in h_{σ (in)}} \\ {\frac{2 \prod_{j = 1}^{n} γ_{σ (ij)}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}}{\prod_{j = 1}^{n} {(2 - γ_{σ (ij)})}^{w {(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}_{j}} + \prod_{j = 1}^{n} {(γ_{σ (ij)})}^{(μ (A_{σ (ij)}) - μ (A_{σ (i, j - 1)}))}}} \end{matrix}$ (7)

to derive the overall values h_i (i = 1, 2, ⋯ , m) of the alternative A_i.

Step 2. Calculate the scores $S ({\tilde{h}}_{i})$ of the hesitant fuzzy value ${\tilde{h}}_{i} (i = 1, 2, \dots, m)$ to rank all the alternatives A_i (i = 1, 2, ⋯ , m) and then to select the best one(s).

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with $S ({\tilde{h}}_{i}) (i = 1, 2, \dots, m)$ .

Step 4. End.

5 Numerical example

Business process change has a long history. Industrial revolution begun in the late 18th century has led people to start thinking about the organization of production processes. In the early 19th century, Henry Ford created their production processes to achieve the dream that the American middle class can own their cars, their success was regarded as the classics of business process change. In the 1990 s, business process reengineering (BPR), known as the “third management revolution” swept globally. Some large and medium-sized enterprises have conducted BPR, with a view to obtain improved performance significantly. In the late 1990 s, the emergence of the Internet, E-mail and Web attracted the enterprises. As a new business model based on computer and network technology, the emergence of e-commerce not only have changed the way that the enterprises sell their products and provide their services, but also have changed the mode of business operation. As a result, fundamental shocks have been brought about to the business processes of the traditional enterprises. Thus, in this section we shall present a numerical example for evaluating the innovation ability of traditional enterprise’s business model for internet transition with hesitant fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible traditional enterprise A_i (i = 1, 2, 3, 4, 5) to select. The experts selects four attribute to evaluate the five possible traditional enterprise: ding172G₁ is the product quality; ding173G₂ is the service; ding174G₃ is the delivery; ding175G₄ is the price. In order to avoid influence each other, the decision makers are required to evaluate the five possible traditional enterprise A_i (i = 1, 2, 3, 4, 5) under the above four attributes in anonymity and the decision matrix H = (h_ij) _4×4 is presented in Table 1, where h_ij (i = 1, 2, 3, 4, j = 1, 2, 3, 4) are in the form of HFEs.

In the following, we utilize the approach developed for evaluating the innovation ability of traditional enterprise’s business model for internet transition with hesitant fuzzy information.

Step 1. Suppose the fuzzy measure of attribute of G_j (j = 1, 2, ⋯ , n) and attribute sets of G as follows: $\begin{matrix} μ (G_{1}) = 0.28, μ (G_{2}) = 0.32, μ (G_{3}) = 0.34 \\ μ (G_{4}) = 0.28, μ (G_{1}, G_{2}) \\ = 0.62, μ (G_{1}, G_{3}) = 0.71 \\ μ (G_{1}, G_{4}) = 0.63, μ (G_{2}, G_{3}) = 0.60 \\ μ (G_{2}, G_{4}) = 0.55, μ (G_{3}, G_{4}) = 0.50 \\ μ (G_{1}, G_{2}, G_{3}) = 0.80, μ (G_{1}, G_{2}, G_{4}) = 0.78 \\ μ (G_{1}, G_{3}, G_{4}) = 0.82, μ (G_{2}, G_{3}, G_{4}) = 0.85 \\ μ (G_{1}, G_{2}, G_{3}, G_{4}) = 1.00 \end{matrix}$

Step 2. The experts use order-inducing variables to represent the complex attitudinal character involving the opinion of different members of the board of directors. The results are shown in Table 2.

Step 3. We utilize the decision information given in Table 1, and the IHFHCG operator to obtain the overall values h_i (i = 1, 2, 3, 4, 5) of the schools A_i (i = 1, 2, 3, 4, 5) and calculate the scores s (h_i) (i = 1, 2, 3, 4, 5) of the overall hesitant fuzzy values h_i (i = 1, 2, 3, 4, 5) of the traditional enterprise A_i: $\begin{matrix} s (h_{1}) = 0.27, s (h_{2}) = 0.12 \\ s (h_{3}) = 0.39, s (h_{4}) = 0.18 \\ s (h_{5}) = 0.23 . \end{matrix}$

Step 3. Rank all the traditional enterprise in accordance with the scores of s (h_i) (i = 1, 2, 3, 4, 5) of the hesitant fuzzy values h_i (i = 1, 2, ⋯ , 5): A₃ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₂, and thus the most desirable traditional enterprise with best innovation ability is A₃.

6 Conclusion

The transportation industry is a basic one in the system of the national economy, the development of the industry has always been the miniature of social, economic and cultural development. With the popularity and development of modern communications and electronic technology, industrial production, commercial mode of operation and consumer model has undergone tremendous changes. These changes objectively request transportation enterprises operating beyond the traditional principles and practices, develop into modern logistics services. According to the definition of marketing given by the Marketing Association in America “Marketing is the process of planning and executing the conception, pricing, promotion, and distribution of ideas, goods, services to create exchange that satisfy individual and organizational goals”, idea, product and service have been included in the area of marketing management. As a new business model based on computer and network technology, the emergence of e-commerce not only have changed the way that the enterprises sell their products and provide their services, but also have changed the mode of business operation. As a result, fundamental shocks have been brought about to the business processes of the traditional enterprises. In this paper, we investigate the multiple attribute decision making with hesitant fuzzy information. Motivated by the induced Choquet ordered averaging operator (I-COA) operator and geometric mean, we develop the induced hesitant fuzzy Hamacher correlated geometric (IHFHCG) operator and then utilize IHFHCG operator to develop the model for multiple attribute decision making with hesitant fuzzy information. Finally, an illustrative example for evaluating the innovation ability of traditional enterprise’s business model for internet transition is given to verify the developed approach. In the future, we shall extend the proposed approaches to other domains [26 –38].

References

Chen

, Xu

Z.S.

and Xia

M.M.

, Interval-valued hesitant preference relations and their applications to group decision making, Knowledge-Based Systems37 (2013), 528–540.

Chen

, Xu

Z.S.

and Xia

M.M.

, Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modelling37(4) (2013), 2197–2211.

Farhadinia

, Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets, Information Sciences240 (2013), 129–144.

Wei

G.W.

, Zhao

X.F.

and Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems46 (2013), 43–53.

Babitha

K.V.

and John

S.J.

, Hesitant fuzzy soft sets, Journal of New Results in Science3 (2013), 98–107.

Yang

X.B.

, Song

X.N.

, Qi

Y.S.

and Yang

J.Y.

, Constructive and axiomatic approaches to hesitant fuzzy rough set, Soft Computing18(6) (2014), 1067–1077.

Feng

, Li

Y.M.

and Leoreanu-Fotea

, Application of level soft sets in decision making based on interval-valued fuzzy soft sets, Computers & Mathematics with Applications60(6) (2010), 1756–1767.

Wang

X.R.

, Zh, , Gao

, Zhao

X.F.

and Wei

G.W.

, Model for evaluating the government archives website’s construction based on the GHFHWD measure with hesitant fuzzy information, International Journal of Digital Content Technology and its Applications5(12) (2011), 418–425.

Z.S.

, Xia

and Chen

, Some hesitant fuzzy aggregation operators with their application in group decision making, Group Decision and Negotiation22(2) (2013), 259–279.

10.

Wei

G.W.

, Zhao

X.F.

and Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems46 (2013), 43–53.

11.

Wei

G.W.

, Hesitant fuzzy prioritized operators and their application to multiple attribute group decision making, Knowledge-Based Systems31 (2012), 176–182.

12.

Wei

G.W.

and Zhao

X.F.

, Induced hesitant interval-valued fuzzy einstein aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems24 (2013), 789–803.

13.

Wei

G.W.

, Zhao

X.F.

, Wang

H.J.

and Lin

, Hesitant fuzzy choquet integral aggregation operators and their applications to multiple attribute decision making, Information: An International Interdisciplinary Journal15(2) (2012), 441–448.

14.

Zhu

, Xu

Z.S.

and Xia

M.M.

, Hesitant fuzzy geometric Bonferroni means, Information Sciences205(1) (2012), 72–85.

15.

Zhang

and Wei

G.W.

, Extension of VIKOR method for decision making problem based on hesitant fuzzy set, Applied Mathematical Modelling37 (2013), 4938–4947.

16.

Meng

F.Y.

, Chen

X.H.

and Zhang

, Multi-attribute decision analysis under a linguistic hesitant fuzzy environment, Information Sciences267 (2014), 287–305.

17.

Z.S.

and Zhang

X.L.

, Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information, Knowledge-Based Systems52 (2013), 53–64.

18.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems25 (2010), 529–539.

19.

Xia

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52(3) (2011), 395–407.

20.

Z.S.

and Xia

, On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligence Systems26(5) (2011), 410–425.

21.

Z.S.

and Xia

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences181(11) (2011), 2128–2138.

22.

Hamachar

, Uber logische verknunpfungenn unssharfer Aussagen undderen Zugenhorige Bewertungsfunktione, in Tral , Klir , Riccardi (Eds.), Progress in Cybernatics and systems research, vol. 3, Hemisphere, Washington DC, (1978), pp. 276–288.

23.

Zhou

, Zhao

and Wei

, Hesitant fuzzy hamacher aggregation operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems26(6) (2014), 2689–2699.

24.

Yager

R.R.

, Induced aggregation operators, Fuzzy Sets and Systems137 (2003), 59–69.

25.

Z.S.

and Da

Q.L.

, An overview of operators for aggregating information, International Journal of Intelligent System18 (2003), 953–969.

26.

Chiclana

, Herrera

, Herrera-Viedma

, The ordered weighted geometric operator: Properties and application, In: Proc of 8th Int Conf on Information Processing and Management of Uncertainty in Knowledge-based Systems, Madrid, (2000), pp. 985–991.

27.

Z.X.

, Chen

M.Y.

, Xia

G.P.

and Wang

, An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making, Expert Systems with Applications38 (2011), 15286–15295.

28.

Wei

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems17(3) (2015), 484–489.

29.

Wei

G.W.

, Wang

J.M.

and Chen

, Potential optimality and robust optimality in multiattribute decision analysis with incomplete information: A comparative study, Decision Support Systems55(3) (2013), 679–684.

30.

Z.S.

, Uncertain Multiple Attribute Decision Making: Methods and Applications, Tsinghua Uuniversity Press, Beijing, China, 2004.

31.

Chen

Y.F.

, Peng

X.D.

, Guan

G.H.

and Jiang

H.D.

, Approaches to multiple attribute decision making based on the correlation coefficient with dual hesitant fuzzy information, Journal of Intelligent and Fuzzy Systems26(5) (2014), 2547–2556.

32.

, Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making, Applied Mathematical Modelling38(2) (2014), 659–666.

33.

Xiao

, Xia

, Gong

and Li

, The trapezoidal fuzzy soft set and its application in MCDM, Applied Mathematical Modelling36(12) (2012), 5844–5855.

34.

Jiang

, Tang

, Chen

, Liu

and Tang

, Interval-valued intuitionistic fuzzy soft sets and their properties, Computers & Mathematics with Applications60(3) (2010), 906–918.

35.

Yang

, Tan

and Meng

C.C.

, The multi-fuzzy soft set and its application in decision making, Applied Mathematical Modelling37(7) (2013), 4915–4923.

36.

Jun

Y.B.

and Park

C.H.

, Applications of soft sets in ideal theory of BCK/BCI-algebras, Information Sciences178(11) (2008), 2466–2475.

37.

Wei

G.W.

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems17(3) (2015), 484–489.

38.

Wei

G.W.

, Wang

J.M.

and Chen

, Potential optimality and robust optimality in multiattribute decision analysis with incomplete information: A comparative study, Decision Support Systems55(3) (2013), 679–684.